of the Lax form (REF ) with rational functions
\(A(x,t)\)  and \(B_i(x,t)\)  satisfying (REF ) and ()
yield nonlinear \(q\) -difference system satisfied by the elements of \(A(x,t)\) , \(B_i(x,t)\) 
as functions of the variables \(t\) 's with the parameters \(\theta \) 's. When \(m=1\)  and \(N=2\) ,
the derived nonlinear \(q\) -difference system is the \(q\) -difference VI equation [1]}. Fortunately,
the dependent variables of \(q\) -\(\mathrm {P_{VI}}\)  are simply written by the elements
of \(A(x,t)\)  and \(B_2(x,t)\)  and moreover they are expressed in terms of
Fourier transforms of \(q\) -conformal blocks, namely, tau functions [2]}.
However, in the general case, nonlinear \(q\) -difference systems derived from compatibility conditions
are complicated and we have not found a way to nicely write down equation satisfied by the elements of \(A(x,t)\)  and \(B_i(x,t)\) . We refer to [3]} where the author
investigated the \(N=2\)  case and obtained a \(q\) -analog of the Garnier system, see also
[4]}, [5]}, [6]}, [7]}. We also refer to [8]} where the authors verified that the Lax form of a generalization \(q\) -\(P_{(n,n)}\)  of \(q\) -\(\mathrm {P_{VI}}\) 
obtained from the Drinfeld–Sokolov hierarchy is transformed to our Lax form (REF )
for \(m=1\)  by \(q\) -Laplace transformations [8]}.
